I've been to Oz 4 times and have never gone with an organized tour.
No need to. It's an easy country to get about on your own. You've
got plenty of time to read the guide books and do your planning.
Once you're there, you may want to take an organized day tour within
an area, but you can plan that from here or there. However, you do
need to plan your interAustralia plane flights in advance because
it's much cheaper from here. Qantas calls it the boomerang pass and
it's a bit flexible should you plan to stay an extra day or leave a
place a day early.

For a 2-2.5 week visit, I'd stick to the eastern coast of Oz as it's
closer to NZ, so less air travel time.
You could probably hit Sydney, Melbourne, Cairns (Great Barrier Reef)
and either the north or south island of NZ. I've only been to the
north island. Auckland is okay for a day, then drive to Rotorua and
spend time exploring that area.
You won't need a car for Sydney or Melbourne and if you go to one of
the resorts in the Great Barrier Reef, you probably don't need one
there either (we've done it with and without). But you do need one
in NZ if you're going to explore the natural sights.

I think the cities I've mentioned are Don't miss. I've yet to go
to Ayres Rock, seems like too much traveling for a huge rock. I've
seen enough photo's of it to last me a lifetime. I'm sure some would
disagree, but that's what makes horse races!